Circuits, Logic, and Games 07.02.10 - 12.02.10

Organizers

Benjamin Rossmann

MIT - Cambridge, USA

Thomas Schwentick

U. Dortmund, DE

Denis Thérien

McGill U. - Montreal, CA

Heribert Vollmer

U. Hannover, DE

Description of the Seminar's Topic

Starting with the seminal paper by Furst, Saxe and Sipser, the last twodecades of the previous century saw an immense interest in the computa-tional model of Boolean circuits. Emerging powerful lower bound techniquespromised progress towards solutions of major open problems in computa-tional complexity theory. Within a very short time, further progress wasmade in papers by Fagin et al., Gurevich and Lewis, HÃ¥stad, Razborov,Smolensky, and Yao, to mention only a few. The just mentioned resultby Furst et al. was obtained independently by Ajtai making use of model-theoretic arguments, and many further lower bounds in complexity have beenobtained afterwards making use of inexpressibility results in logic, very oftenmaking use of model-theoretic games. After a decade of active research inthis direction things slowed down considerably.

During the first seminar on Circuits, Logic, and Games (Nov. 2006,06451), the organizers aimed to bring together researchers from the areasof nite model theory and computational complexity theory, since they feltthat perhaps not all developments in circuit theory and in logic had been ex-plored fully in the context of lower bounds. In fact, the interaction betweenthe areas has nourished a lot in the past 2-3 years, as can be exemplied bythe following lines of research.

Results of Barrington, Straubing and Thérien show that most circuitclasses, if they can be separated at all, can be separated by regular languages- which means that algebraic properties of such languages could be used inlower bound proofs. Recent results prove almost linear upper bounds onthe size of circuits for regular languages in many important constant-depthcircuit classes, implying that an â„¦(n^(1+Îµ) ) lower bound suces to separatesuch classes from NC^1 . Interesting connections to communication complexityhave been obtained in the past two years, showing, e. g., that languageswith bounded multiparty communication complexity can be recognized byprograms over commutative monoids and thus have very small depth circuitcomplexity

While inexpressibility results in finite model theory have been used sincethe 1980s to obtain circuit lower bounds, recent results of Rossman make useof circuit lower bounds to separate different logics : He concluded that thenumber-of-variable hierarchy in rst-order logic over nite ordered structuresis strict by showing that the k -clique-problem (for graphs with n nodes) can-not be solved by constant-depth circuits of size n^(1/4) . The circuit lower boundmakes use of the theory of random graphs and a game-theoretic argument.

Further connections between logic and circuits concern uniformity condi-tions for Boolean circuits : Building on 20 year old results from Immerman etal., Behle and Lange recently proved that in a quite general context, whena circuit based language class is characterized using rst-order descriptivecomplexity, the circuit uniformity conditions spring up in the logic in theform of restrictions on the set of numerical predicates allowed. McKenzie,Thomas, and Vollmer studied so called "extensional uniformity conditions":Intersecting a non-uniform constant-depth circuit class with a uniform classL (e. g., a formal language class) in some contexts results in a circuit classthat can be characterized by rst-order logic with L-numerical predicates.(Intuitively, L-numerical predicates are those predicates denable in rst-order logic with one "oracle call" to a language from L, i. e., more precisely,with one appearance of a generalized quantier for such a language.)

While this in principal points out new ways to separate uniformity con-ditions via logical means, results of Allender, Barrington, and Hesse go inthe opposite direction: They show that for a specic arithmetical problem(division), circuits can be constructed that are uniform under much stricterrequirements than was anticipated before. Again, their proofs make heavyuse of nite model theory.

Employing techniques from nite model theory in form of particularEhrenfeucht-FraÃ¯ssé games, new tools have been developed to obtain lowerbounds for the size of Boolean formulas (that is, circuits with fan-out 1)and rst-order formulas. Adler and Immerman (and, unpublished, Hellaand Väänänen) dened EF games that characterize denability of classesof structures with rst-order formulas of given size. Using these techniques,Immerman and Weis obtained lower bounds for formulas with bounded num-ber of variables and studied the trade-off between formula size and numberof variables. The game based techniques were also applied by Grohe andSchweikardt who initiated a systematic study of different logics.

A further area of investigation is the structural complexity of dynamicalgorithmic problems. There are, so far, no techniques available to prove thata problem does not have AC^0 update complexity. Recent (and forthcoming)work therefore started an investigation of the ne structure of the class ofproblems with AC^0 updates, yielding lower bound results and uncovering(yet another) surprising characterization of the regular languages as thosethat can be maintained with quantier-free formulas.

These results demonstrate the impressive growth of interest and activityin the intersection of nite model theory and Boolean circuit complexity, andas will have become apparent from our above description of recent research,many of these developments rely strongly on game-based methods.

It will be the aim of the workshop to bring together researchers fromthese two areas to further strengthen the mutual fertilization.